OP, the problem with the question
is that, in a way, we have been discouraged from answering this beyond the vaguest terms by the Marxist tradition.
Before Marx, there were plenty of people who would give you a detailed description of what their idea of a communist or socialist society is. They would describe the production, the credit system, the political management, the division of labor, the social function of money, the allocation of resources, the family, the education, everything.
After a while, particularly once several failed "experiments" with isolated communities made it clear that you can't scratch society and recreate everything according to new principles at will, a more historical approach, advocated by the Marxists, has become the dominant one.
This approach doesn't really concern itself with coming up with the specific arrangements and institutions of the communist society, but understanding how the arrangements and institutions of each society come into being in the first place. See, it's something we accept, without much reflection, that the way we live is part manifestation of innate human traits, part the result of a long process of reason and reflection. Marx added a different component to this, by suggesting that the way we organize our production and distribution plays a huge, huge role in shaping our society at large, and that by gaining understanding of the historical developments in this key productive factor, you understand how societies change and evolve through different stages.
To give you an example, if you could ask an early advocate of capitalism what a society ran on these principles would look like, he probably wouldn't be able to tell you shit about our complex financial structure, our political system, our cultural traditions and practices, our economic science, our trade patterns, our education system. But he understood capitalism at its basics: he understood trade, private ownership of productive units, he understood wage labour, and he understood accumulation. Its when this latter combinations grows to dominate more and more aspects of our social life, and the classes that correspond to it (the property owner and the wage labourer) become the dominant classes in society, that all these things we know about "Capitalism" this proto-capitalist doesn't start being figured out, because usually society adapts itself to the way we organize our working life, not the opposite. You have this "base" of capitalist production, or feudal production, or slave production, or primitive communal production, and the structure that surrounds it is shaped by it more than it shapes it.
With this fundamental understanding of how society changes, Marx told people not to bother trying to figure out a fictional society where the new egalitarian values would dominate, he called those socialists "utopian" (not, as we tend to imagine, in a dismissive tone - he deeply respected many of them) and told us, instead, to focus on production. In particular, the production within our reach, that of the working place, suggesting that it can, given certain conditions, be adjusted according to new principles and function as a vehicle to a new society.